## Laws of Motion

# Conservation of Momentum, Equilibrium of a Particle and friction

- When no external forces are acting on the system the total momentum of the system remains constant.
- Total momentum of system before collision is equal to the total momentum of system after collision.
- When two objects exert forces on each other in action reaction, their motions are affected as a pair m
_{1}v_{1}+ m_{2}v_{2}= 0 - Recoil velocity of gun is \tt v_{g} = -\frac{-m_{b} v_{b}}{M_{g}}
- Change in momentum ΔP = mv – mu.
- Resultant momentum \tt \Delta P = \sqrt{P_{1}^{2} + P_{2}^{2} + 2P_{1}P_{2} \cos \theta}
- Change in momentum in opposite direction ΔP = m(v + u).
- A ball hits wall with “u” at an angle “θ” with normal to wall ΔP = 2 mu cos θ.
- Change in momentum along wall = 0.

**Friction:**

- The frictional force acting between two surfaces at rest with respect to each other is called static friction f
_{s}= μ_{s}N. - The frictional force between surfaces in motion is called Kinetic friction f
_{K}= μ_{K}N. - The frictional force when rolls over, the surface of another called Rolling friction f
_{R}= μ_{R}N. - The cofficient of friction μ is f
_{s}> f_{K}> f_{R}μ_{s}> μ_{K}> μ_{R .} - The angle between net contact force and Normal reaction called angle of friction tan θ = fs/N.
- Acceleration of body moving over rough surface \tt a = \frac{F- \mu_k \mu_g}{M}
- Frictional force on a block of mass ‘M’ when a pushing force “F” is applied making an angle ‘θ’ with horizontal F = μ (μg − F sin θ)
- Minimum force acquired to pull \tt F = \frac{- \mu m g}{\cos \theta + \mu \sin \theta}
- Frictional force on a block of mass M when a pushing force ‘F’ is applied making an angle ‘θ’ with horizontal F = μ (μg + F sin θ)
- Minimum force required to pull \tt F = \frac{\mu m g}{\cos \theta - \mu \sin \theta}
- Acceleration of block pressed to vertical wall \tt a = \frac{Mg - f_{k}}{m}
- Block of mass ‘m’ kept on truck moving with a
_{τ}(acceleration) acceleration of block \tt a_{B} = \frac{ma_{\tau} - f_{k}}{m} - Velocity of block \tt v = \sqrt{2 (a_{\tau} - \mu k g) x}
- Time taken to move \tt t = \sqrt{\frac{2x}{\left(a_{\tau} - \mu k g\right)}}
- Acceleration of block of a smooth incline a = g sin θ
- Time taken to reach bottom t = \tt \sqrt{\frac{2 l}{g \sin \theta}}
- Velocity of block at bottom \tt v = \sqrt{2 g l \sin \theta}
- Acceleration of block on rough inclined plane a = g (sin θ – μK cos θ)
- Velocity of block to reach bottom \tt v = \sqrt{2 g l (\sin \theta - \mu k \cos \theta)}
- Time taken to reach bottom \tt t = \sqrt{\frac{2 l}{g (\sin \theta - \mu k \cos \theta)}}
- Force required to move the block up the plane F = mg (sin θ + μK cos θ).
- When the body thrown up with velocity “U” The retarding force F = mg (sin θ + μK cos θ).
- The retardation a = g (sin θ + μK cos θ)
- Time taken to stop \tt t = \sqrt{\frac{2 l}{g (\sin \theta + \mu k \cos \theta)}}
- Distance travelled by body to stop \tt s = \frac{u^{2}}{2 g (\sin \theta + \mu k \cos \theta)}
- Acceleration of system of two Blocks when Force is applied on Bottom block a
_{max}= μs g.

- Maximum force for which blocks move together F
_{max}= μs (M+m)g - Acceleration of Lower block \tt a = \frac{F - \mu k mg}{M}
- Acceleration of system of two blocks when force is applied on top block \tt a = \frac{\mu s mg}{M}

- Maximum force for which both blocks move together \tt F = (M + m) \frac{\mu s mg}{M}
- Acceleration of lower block = \tt \frac{\mu_k mg}{M}
- Acceleration of upper block = \tt \frac{F - \mu_k mg}{m}

### View the Topic in this video From 0:03 To 08:03

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1. According to this law of conservation of linear momentum for a system of particles \overrightarrow{F} = \frac{d\overrightarrow{p}}{dt}

2. If two forces F_{1} & F_{2} act an a particle then they will be in equilibrium \therefore \ \overrightarrow{F_{2}} = -\overrightarrow{F_{1}}

3. Recoil velocity of gun \overrightarrow{v}_{G} = - \frac{m_{B}}{m_{G}}{\overrightarrow{v}_{B}}

4. If 'n' bullets each of mass m are fired per unit time from a machine gun, then the force required to hold the gun \tt = v \left[\frac{dm}{dt}\right] = v (mn) = mnv.

5. Thrust on the rocket : F = -u \frac{dm}{dt}

6. Acceleration of the rocket : \tt a = \frac{u}{m} \frac{dm}{dt} -g

7. Instantaneous velocity of the rocket :\tt v = u \log_{e}\left[\frac{m_{0}}{m}\right] - gt

8. Burnt out speed of the rocket : \tt v_{b} = v_{max} = u\log_{e}\left[\frac{m_{0}}{m_{r}}\right]

9. Weight (W) → w = mg

10. When a lift is at rest or moving with a constant speed, then R = mg

11. When a lift is accelerating upward, then apparent weight *R*_{1} = *m*(*g* + *a*)

12. When a lift is accelerating downward, then apparent weight *R*_{2} = *m*(*g* − *a*)

13. When lift is falling freely under gravity, then *R*_{2} = *m*(*g* − *g*) = 0

14. Minimum force required to move the body up the inclined plane ƒ_{1} = *mg* (sin θ + μ cos θ)

15. Minimum force required to push the body down the inclined plane ƒ_{1} = *mg* (μ cos θ − sin θ)

16.** Two Bodies in Contact:**

Acceleration of the bodies a = \frac{F}{(m_{1} + m_{2})}

Contact force on m_{1} = m_{1}a = \frac{m_{1}F}{(m_{1} + m_{2})}

Contact force on m_{2} = m_{2}a = \frac{m_{2}F}{(m_{1} + m_{2})}

17.** Three Bodies in Contact:**

Acceleration of the bodies = \frac{F}{(m_{1} + m_{2} + m_{3})}

Contact force between m_{1} and m_{2} → F_{1} = \frac{(m_{2} + m_{3})F}{(m_{1} + m_{2} + m_{3})}

Contact force between m_{2} and m_{3} → F_{2} = \frac{m_{3}F}{(m_{1} + m_{2} + m_{3})}

18.** Motion of Two Bodies, One Resting on the Other:**

Acceleration of two bodies

a = \frac{F}{(M + m)}

Pseudo force acting on block *B* due to the accelerated motion ƒ' = ma

19.** Motion of Bodies Connected by strings:**

Acceleration of the system a = \frac{F}{(m_{1} + m_{2} + m_{3})}

Tension in string *T*_{1} = *F*

T_{2} = (m_{2} + m_{3})a = \frac{(m_{2} + m_{3})F}{(m_{1} + m_{2} + m_{3})}

T_{3} = m_{3}a = \frac{m_{3}F}{(m_{1} + m_{2} + m_{3})}

20.** Pulley Mass System: **

When unequal masses m_{1} and m_{2} are suspended from a pulley (m_{1} > m_{2})

a = \frac{(m_{1} - m_{2})}{(m_{1} + m_{2})}g

T = \frac{2m_{1}m_{2}}{(m_{1} + m_{2})}g